MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

While reading about the Burnside problem, I thought of the following question:

 If every proper subgroup of G is finite, does it follow that G is also finite?

Despite extensive searching (and thinking), I am unable to find a solution. (I suspect that the answer is no)

share|cite|improve this question
A Google search for the exact quote "every proper subgroup is finite" helps. – Jonas Meyer Feb 5 '13 at 4:38
For a finitely generated example see Tarski monster. – Benjamin Steinberg Feb 5 '13 at 5:14
I'm fairly sure this question or some slight variation had been asked before with the same answers. – Benjamin Steinberg Feb 5 '13 at 5:20
up vote 18 down vote accepted

No. The direct limit of the cyclic groups of order $p^n$ is infinite, but every proper subgroup of it is finite.

share|cite|improve this answer
I like it. It is the rotation group of the circle of length 1 by elements of Z[1/2]. – Matt Brin Feb 7 '13 at 2:39

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.